Definition: An integer is said to be divisible by an integer , in symbols , if there exists some integer such that .

If b is infinity , then what are the possible a?

I think can be any number. Is it true?

Aug 19th 2009, 07:17 AM

nikhil

according to me

yes. (b/a) should not be undefined.
even infinity/infinity is valid though its a inderminate form as infinity can not be compared with other infinity unless we know its generating function.

Aug 19th 2009, 07:24 AM

BobP

Your definition says that are integers.
Can you reasonably say that is an integer, (which would be necessary to include it within your definition) ?

Aug 19th 2009, 07:34 AM

nikhil

Quote:

Originally Posted by BobP

Your definition says that are integers.
Can you reasonably say that is an integer, (which would be necessary to include it within your definition) ?

well if one is talking about integers then infinity will be an integer
if one is talking about natural numbers then infintity will be an natural number for sure.
it depends upon the function used.
note:
if there are two person A and B. they are talking about natural numbers, both of them then think about infinity then you will never be able to find if
(infinity suggested by A)/(infinity suggested by B)>1 or <1 this will be totally indeterminate.

Aug 19th 2009, 09:18 AM

Bruno J.

Quote:

Originally Posted by nikhil

yes. (b/a) should not be undefined.
even infinity/infinity is valid though its a inderminate form as infinity can not be compared with other infinity unless we know its generating function.

What do generating functions have to do with this? Perhaps you mean rate of convergence and such.

Quote:

well if one is talking about integers then infinity will be an integer
if one is talking about natural numbers then infintity will be an natural number for sure.
it depends upon the function used.
note:
if there are two person A and B. they are talking about natural numbers, both of them then think about infinity then you will never be able to find if
(infinity suggested by A)/(infinity suggested by B)>1 or <1 this will be totally indeterminate.

You have it all wrong. "Infinity" is not a natural number, not a real number, not a complex number. For instance, take the Peano axioms as a foundation for arithmetic; then if we must have and .. But you will agree that both of these are equal to hence they cannot be natural numbers because the successor function is injective (it's one of the axioms).

In other words, if you define as a natural number you end up with things such as . The cancellation laws for addition are easily derived from the Peano axioms and they do not hold for "infinity".

Aug 19th 2009, 04:04 PM

Bingk

Assuming that b can be infinity, and a is any integer not equal to zero, consider what integer c multiplied to a would equal infinity? If you multiply any two integers, no matter how "large", you'll still get an integer which might be extremely "large" but still not infinity.
So, to answer your question, a can't be any number ...

Regarding infinity:

Wikipedia: "Infinity (symbolically represented by ∞) refers to several distinct concepts – usually linked to the idea of "without end"".
i.e. Infinity is not a number, it's an idea of being "endless".

Aug 20th 2009, 12:58 AM

mr fantastic

Quote:

Originally Posted by Bingk

Assuming that b can be infinity, and a is any integer not equal to zero, consider what integer c multiplied to a would equal infinity? If you multiply any two integers, no matter how "large", you'll still get an integer which might be extremely "large" but still not infinity.
So, to answer your question, a can't be any number ...

Regarding infinity:

Wikipedia: "Infinity (symbolically represented by ∞) refers to several distinct concepts – usually linked to the idea of "without end"".
i.e. Infinity is not a number, it's an idea of being "endless".